Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Experimentally Testable Noncontextuality Inequalities Via Fourier-Motzkin Elimination

Anirudh Krishna

Published Sep 29, 2015• 3 citations
OpenAlex

Abstract

Generalized noncontextuality as defined by Spekkens is an attempt to make precise the distinction between classical and quantum theories. It is a restriction on models used to reproduce the predictions of quantum mechanics. There has been considerable progress recently in deriving experimentally robust noncontextuality inequalities. The violation of these inequalities is a sufficient condition for an experiment to not admit a generalized noncontextual model. \nIn this thesis, we present an algorithm to automate the derivation of noncontextuality inequalities. At the heart of this algorithm is a technique called Fourier Motzkin elimination (abbrev. FM elimination). \nAfter a brief overview of the generalized notion of contextuality and FM elimination, we proceed to demon- strate how this algorithm works by using it to derive noncontextuality inequalities for a number of examples. \nBelinfante’s construction, presented first for the sake of pedagogy, is based on a simple proof by Belinfante to demonstrate the invalidity of von Neumann’s notion of noncontextuality. We provide a quantum realization on a single qubit that can violate this inequality. \nWe then go on to discuss paradigmatic proofs of noncontextuality such as the Peres-Mermin square, Mermin’s Star and the 18 vector construction and use them to derive robust noncontextuality inequalities using the algorithm. \nWe also show how one can generalize the noncontextuality inequalities obtained for the Peres-Mermin square to systems with continuous variables.

Related papers

Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the Peres–Mermin square

Anirudh Krishna, Robert W. Spekkens, Elie Wolfe

When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen–Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen–Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres–Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres–Mermin square.

arXiv:1704.01153•New Journal of Physics•Oct 5, 2017•31 citations

Noncontextuality inequalities for prepare-transform-measure scenarios

David Schmid, Roberto D. Baldijão, John H. Selby +2 more

We provide the first systematic technique for deriving witnesses of contextuality in prepare-transform-measure scenarios. More specifically, we show how linear quantifier elimination can be used to compute a polytope of correlations consistent with generalized noncontextuality in such scenarios. This polytope is specified as a set of noncontextuality inequalities that are necessary and sufficient conditions for observed data in the scenario to admit of a classical explanation relative to any linear operational identities, if one ignores some constraints from diagram preservation. While including these latter constraints generally leads to tighter inequalities, it seems that nonlinear quantifier elimination would be required to systematically include them. We also provide a linear program which can certify the nonclassicality of a set of numerical data arising in a prepare-transform-measure experiment. We apply our results to get a robust noncontextuality inequality for transformations that can be violated within the stabilizer subtheory. Finally, we give a simple algorithm for computing all the linear operational identities holding among a given set of states, of transformations, or of measurements.

arXiv:2407.09624•Quantum•Oct 27, 2026•2 citations

The robust noncontextuality inequalities in the simplest scenario

Matthew F. Pusey

Contextuality is the leading notion of non-classicality for a single system. However, experimental demonstrations of contextuality have generally been limited by conceptual difficulties. Here I focus on the simplest non-trivial case, four preparations and two tomographically complete binary measurements. Exploiting a subtle connection to the CHSH scenario gives eight non-linear inequalities which are together necessary and sufficient for the experimental statistics to admit a preparation noncontextual model. No fixed operational equivalences are required. Violating such an inequality is therefore arguably the most direct route to an experimental refutation of noncontextuality.

•Jun 12, 2015•8 citations

Experimental generalized contextuality with single-photon qubits

Xiang Zhan, Eric G. Cavalcanti, Jian Li +4 more

Contextuality is a phenomenon at the heart of the quantum mechanical departure from classical behaviour, and has been recently identified as a resource in quantum computation. Experimental demonstration of contextuality is thus an important goal. The traditional form of contextuality -- as violation of a Kochen-Specker inequality -- requires a quantum system with at least three levels, and the status of the assumption of determinism used in deriving those inequalities has been controversial. By considering `unsharp' observables, Liang, Spekkens and Wiseman (LSW) derived an inequality for generalized noncontextual models that doesn't assume determinism, and applies already to a qubit. We experimentally implement the LSW test using the polarization states of a heralded single photon and three unsharp binary measurements. We violate the LSW inequality by more than 16 standard deviations, thus showing that our results cannot be reproduced by a noncontextual subset of quantum theory.

arXiv:1505.06825•Optica•Aug 8, 2017•31 citations

All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences

David Schmid, Robert W. Spekkens, Elie Wolfe

Within the framework of generalized noncontextuality, we introduce a general technique for systematically deriving noncontextuality inequalities for any experiment involving finitely many preparations and finitely many measurements, each of which has a finite number of outcomes. Given any fixed sets of operational equivalences among the preparations and among the measurements as input, the algorithm returns a set of noncontextuality inequalities whose satisfaction is necessary and sufficient for a set of operational data to admit of a noncontextual model. Additionally, we show that the space of noncontextual data tables always defines a polytope. Finally, we provide a computationally efficient means for testing whether any set of numerical data admits of a noncontextual model, with respect to any fixed operational equivalences. Together, these techniques provide complete methods for characterizing arbitrary noncontextuality scenarios, both in theory and in practice. Because a quantum prepare-and-measure experiment admits of a noncontextual model if and only if it admits of a positive quasiprobability representation, our techniques also determine the necessary and sufficient conditions for the existence of such a representation.

arXiv:1710.08434•Physical Review A•Jun 5, 2018•58 citations

From the Kochen-Specker Theorem to Noncontextuality Inequalities without Assuming Determinism

Ravi Kunjwal, Robert W. Spekkens

The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.

arXiv:1506.04150•Physical Review Letters•Sep 9, 2015•87 citations
Generalized noncontextuality as defined by Spekkens is an attempt to make precise the distinction between classical and quantum theories. It is a restriction on models used to reproduce the predictions of quantum mechanics. There has been considerable progress recently in deriving experimentally robust noncontextuality inequalities. The violation of these inequalities is a sufficient condition for an experiment to not admit a generalized noncontextual model. \nIn this thesis, we present an algorithm to automate the derivation of noncontextuality inequalities. At the heart of this algorithm is a technique called Fourier Motzkin elimination (abbrev. FM elimination). \nAfter a brief overview of the generalized notion of contextuality and FM elimination, we proceed to demon- strate how this algorithm works by using it to derive noncontextuality inequalities for a number of examples. \nBelinfante’s construction, presented first for the sake of pedagogy, is based on a simple proof by Belinfante to demonstrate the invalidity of von Neumann’s notion of noncontextuality. We provide a quantum realization on a single qubit that can violate this inequality. \nWe then go on to discuss paradigmatic proofs of noncontextuality such as the Peres-Mermin square, Mermin’s Star and the 18 vector construction and use them to derive robust noncontextuality inequalities using the algorithm. \nWe also show how one can generalize the noncontextuality inequalities obtained for the Peres-Mermin square to systems with continuous variables.
When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen–Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen–Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres–Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres–Mermin square.
We provide the first systematic technique for deriving witnesses of contextuality in prepare-transform-measure scenarios. More specifically, we show how linear quantifier elimination can be used to compute a polytope of correlations consistent with generalized noncontextuality in such scenarios. This polytope is specified as a set of noncontextuality inequalities that are necessary and sufficient conditions for observed data in the scenario to admit of a classical explanation relative to any linear operational identities, if one ignores some constraints from diagram preservation. While including these latter constraints generally leads to tighter inequalities, it seems that nonlinear quantifier elimination would be required to systematically include them. We also provide a linear program which can certify the nonclassicality of a set of numerical data arising in a prepare-transform-measure experiment. We apply our results to get a robust noncontextuality inequality for transformations that can be violated within the stabilizer subtheory. Finally, we give a simple algorithm for computing all the linear operational identities holding among a given set of states, of transformations, or of measurements.
Contextuality is the leading notion of non-classicality for a single system. However, experimental demonstrations of contextuality have generally been limited by conceptual difficulties. Here I focus on the simplest non-trivial case, four preparations and two tomographically complete binary measurements. Exploiting a subtle connection to the CHSH scenario gives eight non-linear inequalities which are together necessary and sufficient for the experimental statistics to admit a preparation noncontextual model. No fixed operational equivalences are required. Violating such an inequality is therefore arguably the most direct route to an experimental refutation of noncontextuality.
Contextuality is a phenomenon at the heart of the quantum mechanical departure from classical behaviour, and has been recently identified as a resource in quantum computation. Experimental demonstration of contextuality is thus an important goal. The traditional form of contextuality -- as violation of a Kochen-Specker inequality -- requires a quantum system with at least three levels, and the status of the assumption of determinism used in deriving those inequalities has been controversial. By considering `unsharp' observables, Liang, Spekkens and Wiseman (LSW) derived an inequality for generalized noncontextual models that doesn't assume determinism, and applies already to a qubit. We experimentally implement the LSW test using the polarization states of a heralded single photon and three unsharp binary measurements. We violate the LSW inequality by more than 16 standard deviations, thus showing that our results cannot be reproduced by a noncontextual subset of quantum theory.
Within the framework of generalized noncontextuality, we introduce a general technique for systematically deriving noncontextuality inequalities for any experiment involving finitely many preparations and finitely many measurements, each of which has a finite number of outcomes. Given any fixed sets of operational equivalences among the preparations and among the measurements as input, the algorithm returns a set of noncontextuality inequalities whose satisfaction is necessary and sufficient for a set of operational data to admit of a noncontextual model. Additionally, we show that the space of noncontextual data tables always defines a polytope. Finally, we provide a computationally efficient means for testing whether any set of numerical data admits of a noncontextual model, with respect to any fixed operational equivalences. Together, these techniques provide complete methods for characterizing arbitrary noncontextuality scenarios, both in theory and in practice. Because a quantum prepare-and-measure experiment admits of a noncontextual model if and only if it admits of a positive quasiprobability representation, our techniques also determine the necessary and sufficient conditions for the existence of such a representation.
The Kochen-Specker theorem demonstrates that it is not possible to reproduce the predictions of quantum theory in terms of a hidden variable model where the hidden variables assign a value to every projector deterministically and noncontextually. A noncontextual value assignment to a projector is one that does not depend on which other projectors-the context-are measured together with it. Using a generalization of the notion of noncontextuality that applies to both measurements and preparations, we propose a scheme for deriving inequalities that test whether a given set of experimental statistics is consistent with a noncontextual model. Unlike previous inequalities inspired by the Kochen-Specker theorem, we do not assume that the value assignments are deterministic and therefore in the face of a violation of our inequality, the possibility of salvaging noncontextuality by abandoning determinism is no longer an option. Our approach is operational in the sense that it does not presume quantum theory: a violation of our inequality implies the impossibility of a noncontextual model for any operational theory that can account for the experimental observations, including any successor to quantum theory.